# Properties

 Label 2160.4.a.bh Level $2160$ Weight $4$ Character orbit 2160.a Self dual yes Analytic conductor $127.444$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.444125612$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.1765.1 Defining polynomial: $$x^{3} - x^{2} - 11x + 16$$ x^3 - x^2 - 11*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + \beta_1 q^{7}+O(q^{10})$$ q - 5 * q^5 + b1 * q^7 $$q - 5 q^{5} + \beta_1 q^{7} + (2 \beta_{2} - \beta_1 - 9) q^{11} + ( - \beta_{2} + 2 \beta_1 - 1) q^{13} + (5 \beta_{2} - 3 \beta_1 + 5) q^{17} + ( - 3 \beta_{2} - 3 \beta_1 + 26) q^{19} + ( - 6 \beta_{2} + \beta_1 - 35) q^{23} + 25 q^{25} + ( - 5 \beta_{2} + \beta_1 + 39) q^{29} + ( - 8 \beta_{2} - \beta_1 + 69) q^{31} - 5 \beta_1 q^{35} + ( - 19 \beta_{2} + 19 \beta_1 - 40) q^{37} + ( - 3 \beta_{2} - 16 \beta_1 + 100) q^{41} + (10 \beta_{2} + 3 \beta_1 + 161) q^{43} + (19 \beta_{2} - 17 \beta_1 - 101) q^{47} + (12 \beta_{2} - 13 \beta_1 - 5) q^{49} + ( - 11 \beta_{2} + 8 \beta_1 - 164) q^{53} + ( - 10 \beta_{2} + 5 \beta_1 + 45) q^{55} + (3 \beta_{2} - 30 \beta_1 - 80) q^{59} + (20 \beta_{2} - 37 \beta_1 - 148) q^{61} + (5 \beta_{2} - 10 \beta_1 + 5) q^{65} + (36 \beta_{2} + \beta_1 + 174) q^{67} + ( - 17 \beta_{2} + 52 \beta_1 + 56) q^{71} + (27 \beta_{2} - 11 \beta_1 - 292) q^{73} + (14 \beta_{2} + 4 \beta_1 - 50) q^{77} + (3 \beta_{2} + 10 \beta_1 + 701) q^{79} + ( - 53 \beta_{2} + 26 \beta_1 + 14) q^{83} + ( - 25 \beta_{2} + 15 \beta_1 - 25) q^{85} + (13 \beta_{2} + 30 \beta_1 - 756) q^{89} + (11 \beta_{2} - 27 \beta_1 + 532) q^{91} + (15 \beta_{2} + 15 \beta_1 - 130) q^{95} + (29 \beta_{2} - 3 \beta_1 - 464) q^{97}+O(q^{100})$$ q - 5 * q^5 + b1 * q^7 + (2*b2 - b1 - 9) * q^11 + (-b2 + 2*b1 - 1) * q^13 + (5*b2 - 3*b1 + 5) * q^17 + (-3*b2 - 3*b1 + 26) * q^19 + (-6*b2 + b1 - 35) * q^23 + 25 * q^25 + (-5*b2 + b1 + 39) * q^29 + (-8*b2 - b1 + 69) * q^31 - 5*b1 * q^35 + (-19*b2 + 19*b1 - 40) * q^37 + (-3*b2 - 16*b1 + 100) * q^41 + (10*b2 + 3*b1 + 161) * q^43 + (19*b2 - 17*b1 - 101) * q^47 + (12*b2 - 13*b1 - 5) * q^49 + (-11*b2 + 8*b1 - 164) * q^53 + (-10*b2 + 5*b1 + 45) * q^55 + (3*b2 - 30*b1 - 80) * q^59 + (20*b2 - 37*b1 - 148) * q^61 + (5*b2 - 10*b1 + 5) * q^65 + (36*b2 + b1 + 174) * q^67 + (-17*b2 + 52*b1 + 56) * q^71 + (27*b2 - 11*b1 - 292) * q^73 + (14*b2 + 4*b1 - 50) * q^77 + (3*b2 + 10*b1 + 701) * q^79 + (-53*b2 + 26*b1 + 14) * q^83 + (-25*b2 + 15*b1 - 25) * q^85 + (13*b2 + 30*b1 - 756) * q^89 + (11*b2 - 27*b1 + 532) * q^91 + (15*b2 + 15*b1 - 130) * q^95 + (29*b2 - 3*b1 - 464) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5}+O(q^{10})$$ 3 * q - 15 * q^5 $$3 q - 15 q^{5} - 27 q^{11} - 3 q^{13} + 15 q^{17} + 78 q^{19} - 105 q^{23} + 75 q^{25} + 117 q^{29} + 207 q^{31} - 120 q^{37} + 300 q^{41} + 483 q^{43} - 303 q^{47} - 15 q^{49} - 492 q^{53} + 135 q^{55} - 240 q^{59} - 444 q^{61} + 15 q^{65} + 522 q^{67} + 168 q^{71} - 876 q^{73} - 150 q^{77} + 2103 q^{79} + 42 q^{83} - 75 q^{85} - 2268 q^{89} + 1596 q^{91} - 390 q^{95} - 1392 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 - 27 * q^11 - 3 * q^13 + 15 * q^17 + 78 * q^19 - 105 * q^23 + 75 * q^25 + 117 * q^29 + 207 * q^31 - 120 * q^37 + 300 * q^41 + 483 * q^43 - 303 * q^47 - 15 * q^49 - 492 * q^53 + 135 * q^55 - 240 * q^59 - 444 * q^61 + 15 * q^65 + 522 * q^67 + 168 * q^71 - 876 * q^73 - 150 * q^77 + 2103 * q^79 + 42 * q^83 - 75 * q^85 - 2268 * q^89 + 1596 * q^91 - 390 * q^95 - 1392 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 11x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$-3\nu^{2} + 3\nu + 22$$ -3*v^2 + 3*v + 22 $$\beta_{2}$$ $$=$$ $$-6\nu^{2} - 6\nu + 48$$ -6*v^2 - 6*v + 48
 $$\nu$$ $$=$$ $$( -\beta_{2} + 2\beta _1 + 4 ) / 12$$ (-b2 + 2*b1 + 4) / 12 $$\nu^{2}$$ $$=$$ $$( -\beta_{2} - 2\beta _1 + 92 ) / 12$$ (-b2 - 2*b1 + 92) / 12

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.48079 2.89055 1.59024
0 0 0 −5.00000 0 −24.7900 0 0 0
1.2 0 0 0 −5.00000 0 5.60585 0 0 0
1.3 0 0 0 −5.00000 0 19.1841 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.4.a.bh 3
3.b odd 2 1 2160.4.a.bp 3
4.b odd 2 1 1080.4.a.f 3
12.b even 2 1 1080.4.a.l yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.f 3 4.b odd 2 1
1080.4.a.l yes 3 12.b even 2 1
2160.4.a.bh 3 1.a even 1 1 trivial
2160.4.a.bp 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}^{3} - 507T_{7} + 2666$$ T7^3 - 507*T7 + 2666 $$T_{11}^{3} + 27T_{11}^{2} - 1272T_{11} + 8044$$ T11^3 + 27*T11^2 - 1272*T11 + 8044

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$(T + 5)^{3}$$
$7$ $$T^{3} - 507T + 2666$$
$11$ $$T^{3} + 27 T^{2} - 1272 T + 8044$$
$13$ $$T^{3} + 3 T^{2} - 1629 T + 19553$$
$17$ $$T^{3} - 15 T^{2} - 9708 T + 420688$$
$19$ $$T^{3} - 78 T^{2} - 10635 T + 766504$$
$23$ $$T^{3} + 105 T^{2} - 11088 T - 502268$$
$29$ $$T^{3} - 117 T^{2} - 5484 T + 275060$$
$31$ $$T^{3} - 207 T^{2} - 19632 T + 3717376$$
$37$ $$T^{3} + 120 T^{2} + \cdots - 22578642$$
$41$ $$T^{3} - 300 T^{2} - 124740 T + 9666000$$
$43$ $$T^{3} - 483 T^{2} + 13440 T + 370864$$
$47$ $$T^{3} + 303 T^{2} - 145332 T + 2148016$$
$53$ $$T^{3} + 492 T^{2} + 29628 T - 8122896$$
$59$ $$T^{3} + 240 T^{2} + \cdots - 117125056$$
$61$ $$T^{3} + 444 T^{2} + \cdots - 202420030$$
$67$ $$T^{3} - 522 T^{2} - 531759 T + 6471308$$
$71$ $$T^{3} - 168 T^{2} + \cdots + 521430912$$
$73$ $$T^{3} + 876 T^{2} + \cdots - 13506642$$
$79$ $$T^{3} - 2103 T^{2} + \cdots - 297054405$$
$83$ $$T^{3} - 42 T^{2} + \cdots - 368484664$$
$89$ $$T^{3} + 2268 T^{2} + \cdots - 159259392$$
$97$ $$T^{3} + 1392 T^{2} + \cdots - 80387830$$